[Paper Review] Rank Two Non-Abelian Zeta and Its Zeros

Lin Weng|ArXiv.org|Dec 1, 2004

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper establishes a novel connection between rank two non-abelian zeta functions and classical Dedekind zeta functions via the Rankin-Selberg and Zagier method, proving that all non-trivial zeros of these zeta functions for number fields lie on the critical line ℜ(s) = 1/2, thereby confirming a generalized Riemann Hypothesis for this class of L-functions through geometric and analytic number theory techniques.

ABSTRACT

In this paper, we first reveal an intrinsic relation between non-abelian zeta functions and Epstein zeta functions for algebraic number fields. Then, we expose a fundamental relation between stability of lattices and distance to cusps. Next, using these two relations, we explicitly express rank two zeta functions in terms of the well-known Dedekind zeta functions. Finally, based on such an expression, we show that all zeros of rank two non-abelian zeta functions are entirely sitting on the critical line whose real part equals to 1/2. This is an integrated part of our Geo-Arithmetic Program.

Motivation & Objective

• To establish a fundamental link between non-abelian zeta functions and Epstein zeta functions via Mellin transforms and lattice stability.
• To define non-abelian zeta functions for number fields as a natural generalization of Dedekind zeta functions.
• To unify algebraic and analytic number theory by expressing rank two non-abelian zeta functions explicitly in terms of Dedekind zeta functions.
• To prove the generalized Riemann Hypothesis for rank two non-abelian zeta functions using symmetry and functional equation techniques.
• To clarify the intrinsic relation between lattice stability and distance to cusps in the upper half-space model

Proposed method

• Utilizes the Rankin-Selberg method and Zagier’s technique to derive an explicit formula for rank two non-abelian zeta functions in terms of Dedekind zeta functions.
• Applies the Mellin transform to relate non-abelian zeta functions to Epstein zeta functions associated with lattices over number fields.
• Constructs a fundamental domain for the action of the special automorphism group on the upper half-space using normalized Siegel-type distances to cusps.
• Establishes a correspondence between cusp stabilizers and ideal classes, enabling a geometric interpretation of lattice isometry classes.
• Employs Fourier expansion techniques for Epstein zeta functions to analyze their analytic behavior and functional structure.
• Uses the functional equation and product representation of the Riemann ξ-function to extend the argument to general number fields

Experimental results

Research questions

• RQ1 Can non-abelian zeta functions for number fields be explicitly expressed in terms of classical Dedekind zeta functions?
• RQ2 What is the geometric and arithmetic significance of the distance from a lattice to its cusps in the context of stability?
• RQ3 Does the generalized Riemann Hypothesis hold for rank two non-abelian zeta functions over arbitrary number fields?
• RQ4 How does the structure of the moduli space of rank two lattices relate to the action of SL(2,𝒪_K) and unit groups?
• RQ5 What role does the symmetry of the functional equation play in ensuring zeros lie on the critical line?

Key findings

• All non-trivial zeros of the rank two non-abelian zeta function for the rational numbers lie on the critical line ℜ(s) = 1/2.
• For any number field K, the rank two non-abelian zeta function ξ_{K,2}(s) is expressible as a linear combination of Dedekind zeta functions via the Rankin-Selberg method.
• The explicit formula ξ_{K,2}(s) = [ξ_K(2s)/(s−1)]Δ_K^{s−1} − [ξ_K(2s−1)/s]Δ_K^{−s} holds up to a constant depending only on K.
• The proof of the Riemann Hypothesis for rank two zeta functions relies on the fact that s(s−1)ξ_K(s) is an entire function of order one.
• The argument extends from ℚ to any number field K due to the uniformity of the functional equation and the condition Δ_K ≥ 1.
• The method fails in the limit T→∞ for the family ξ_{ℚ,2}^T(s), showing that the Riemann Hypothesis is not preserved under such limits